Saturday, 31 August 2013

Equilibrium in the Champions League

The UEFA Champions League draw took place on Thursday. This wasn't for the knock-out stages, touched on in a previous Massive Blog post, but for the groups.

To
create some kind of balance across the 8 groups, UEFA rank all the teams
according to their club coefficient system. Much has been written about the
system, such as this, with Chris Bevan saying things like “it is likely to
preserve the status quo” and it “is stacked in the favour of teams from
stronger nations”.

Whatever you think about the system, it rewards teams over a
5 year period for winning and drawing games in Europe and for progressing to
later rounds of European competitions. You might think it should weight more
recent results more heavily or reward wins against stronger teams more highly,
both of which FIFA’s world country ranking does. However, FIFA’s system is
often criticised as well, so it would seem that international football governing
bodies just can’t win.

Anyway, I’m here to talk about the balance of the groups. At
the moment, they rank the 32 teams and then split them into 4 pots: the best 8,
9th-16th best, 17th to 24th best
and the worst 8. Then, to make the groups balanced, each group contains one
team from each pot.

Looking for more balance

This only provides a very sketchy level of balance, however,
as 8 teams are lumped together in one pot, meaning that Barca, ranked first,
are treated just the same as Benfica, ranked eighth, and so on.

If we go into more detail on the ranking, and, instead of just
splitting the teams into 4 pots, give each team points according to its
ranking, from 1 for Barca down to 32 for Real Sociedad, then we can use these
rankings as a basis for finding more balance.

All 32 teams together have (32+1)*16 = 528 points between
them. For balance then, each group should have 528/8 = 66 points. And we can
still maintain the pots, so that each group also has someone from each of the 4
pots.

To make these groups, the most obvious way would be to put
the best teams from pot 1 and 2 with the worst from pot 3 and 4, giving you
teams 1,9,24 and 32. You could then work down pots 1 and 2 and up pots 3 and 4,
so the next group would have teams 2,10,23,31 and so on. This would give you 8
groups of 66 total points.

However, this very simple method falls at the first hurdle
because the first group would be Barca, Atletico Madrid, Leverkusen and Real
Sociedad, three of which are Spanish, which isn’t allowed.

Excel time

I threw a spreadsheet together and did some trial and error
and you can get balanced groups which also respect the rule of having maximum
one team from each national federation in each group. One way it works is (ranking from 1 to 32 given next to each team):

Group A

Group B

Man
U

5

Real

4

Marseilles

13

Schalke

12

Galatasaray

23

Olympiakos

22

Copenhagen

25

Celtic

28

Group C

Group D

Benfica

8

Bayern

2

PSG

15

CSKA

14

Zenit

17

Man
City

18

Napoli

26

Sociedad

32

Group E

Group F

Chelsea

3

Arsenal

6

Juventus

16

Atletico

9

Dortmund

20

Basel

21

Anderlecht

27

Plzen

30

Group G

Group H

Porto

7

Barcelona

1

Milan

11

Donetsk

10

Ajax

19

Leverkusen

24

Bucharest

29

Vienna

31

In all likelihood, this isn’t the only way the groups could
be compiled in a fully balanced way, but it is one way which serves as an
example and as proof that it can be done. If you wanted, you could almost
certainly let a computer randomly compile balanced groups from all the possible
permutations.

Now, given balanced groups, the average strength of your
opponents (as defined by the ranking points) is always (66 – n)/3, where n is
your ranking. For example, Chelsea’s opponents have an average strength
(66-3)/3 = 21, whereas Arsenal’s have an average (66-6)/3 = 20.

This is slightly obvious, but it serves to demonstrate that
Arsenal would, if all were balanced, have slightly higher-ranked opponents than
Chelsea. And they shouldn’t complain about it because it’s their fault for
being worse-ranked than Chelsea.

A look at the actual
groups

Now that we’ve established that balanced groups are possible
and what they could (in one example) look like, let’s take a look at the actual
groups:

Group A

Group B

Man
U

5

Real

4

Donetsk

10

Juventus

16

Leverkusen

24

Galatasaray

23

Sociedad

32

Copenhagen

25

Total

71

Total

68

Group C

Group D

Benfica

8

Bayern

2

PSG

15

CSKA

14

Olympiakos

22

Man
City

18

Anderlecht

27

Plzen

30

Total

72

Total

64

Group E

Group F

Chelsea

3

Arsenal

6

Schalke

12

Marseilles

13

Basel

21

Dortmund

20

Bucharest

29

Napoli

26

Total

65

Total

65

Group G

Group H

Porto

7

Barcelona

1

Atletico

9

Milan

11

Zenit

17

Ajax

19

Vienna

31

Celtic

28

Total

64

Total

59

As better-ranked teams have a lower number ranking, the lower
the total, the stronger the group. Going by this, the strongest group turns out
as Group H, with a total of 59, and the weakest as Group C, with a total of 72.

Interestingly, in spite of articles like this one, saying
things like “Arsenal have a tricky start” or Phil McNulty saying things like “Arsenal
were given the toughest test” and “Jose Mourinho may have silently celebrated
how the overblown ceremony treated Chelsea”, the table shows that both groups
are equally strong.

This means that Arsenal’s opponents (total ranking 59) are
slightly stronger than Chelsea’s (62), but this serves to maintain the balance,
as Arsenal are worse ranked. Man City’s opponents are obviously significantly
stronger (44), as Man City themselves are only in the 3rd pot.

Getting lucky

Clearly, if all groups were balanced then no team would be
luckier than any other team. To measure the luck of a team, I therefore propose
comparing the strength of their group opponents with the strength of their
opponents in a balanced group.

We already have the formula for a team’s opponents’ average strength
in a balanced group: (66 – n)/3

For the average strength of a team’s opponents in an actual
group, we just replace the 66 with g, the group total: (g-n)/3.

The difference between these two totals then gives us the
luck (l) of the team.

l = ((g-n)/3) – ((66-n)/3)

This can be simplified a bit down to:

3l = g-n – (66-n)

3l = g – n – 66 +
n

3l = g – 66

l = (g-66)/3

Doing it this way round also means that a team playing
lower-ranked opponents will have positive luck, which is nice and intuitive.

Interestingly, perhaps, this shows us that a team’s own rank
(n) has no impact on its luck, as there is no (n) in the formula. This means
that all teams in any group are all as lucky as each other, which is nice.

Luck of the English

So how does it look for the English teams?

Chelsea and Arsenal both have:

(65-66)/3 = -1/3 = -0.33

Man Utd have:

(71-66)/3 = 5/3 = +1.67

and Man City have:

(64-66)/3 = -2/3 = -0.67

This suggests that Man Utd are the only English side which
have really been lucky, Man City have had the worst luck and Chelsea and
Arsenal have both been equally (and only slightly) unlucky.

This would appear to contradict McNulty’s suggestions that “United
and Moyes have not been handed an easy group” and that Arsenal have gotten "on the wrong end of the Champions League draw”.

Luck of the Scottish

If we look at Celtic, they have a luck score of:

(59-66)/3 = -7/3 or -2.33

Add that to the fact that they’re the worst team in their
group, and you can’t really argue with their manager Neil Lennon’s statement
that: “in terms of football it doesn't come any harder”. The worst luck theoretically
possible for Celtic would be -3.33 but -2.33 is certainly the unluckiest of all
groups in the actual draw.

Given what I said earlier about all group members having the
same luck, Celtic’s opponents in Group H, Barca, Milan and Ajax, have also had
just as rotten luck as the Scottish champions. This may be of some consolation
to them.

To sum up, whether or not you like the way the clubs are
seeded, whether or not you agree with the system of club coefficient
calculation and whether or not you’d like to see more balance in the draw, I
think we can say with some certainty that Lennon probably pays more attention than
McNulty to maths.

Coming soon: Tune in
to the next blog post for a look at the balance of the groups using the club
coefficient points as a basis, not just a simple 1 to 32 ranking.

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